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PreprintarXiv licencearXiv

A flat geodesic with a periodic asymptote

  • Hongda Qiu
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Abstract

We construct a closed surface of genus $2$ with nonpositive Gaussian curvature and a complete nonperiodic flat geodesic asymptotic to a flat closed geodesic. The metric is of class C3,1/3 and its curvature is of class C1,1/3, but not C2. Thus the periodic-asymptote obstruction for smooth surfaces does not extend to C1 Gaussian curvature.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026flat,
  title = {{A flat geodesic with a periodic asymptote}},
  author = {Hongda Qiu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30707},
  eprint = {2610.30707},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30707},
}
RIS
TY  - UNPB
TI  - A flat geodesic with a periodic asymptote
AU  - Hongda Qiu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30707
UR  - https://arxiv.org/abs/2610.30707
AB  - We construct a closed surface of genus $2$ with nonpositive Gaussian curvature and a complete nonperiodic flat geodesic asymptotic to a flat closed geodesic. The metric is of class C^3,1/3 and its curvature is of class C^1,1/3, but not C^2. Thus the periodic-asymptote obstruction for smooth surfaces does not extend to C^1 Gaussian curvature.
ER  -